【作者】 李娟；

【导师】 宋永忠； 朱怀平；

【作者基本信息】 南京师范大学 ， 计算数学， 2017， 博士

【摘要】 浅水湖泊是人类生存环境的重要组成部分,它在社会经济可持续发展中发挥着重要作用。近年来,由于全球气候变化、以及湖泊周边地区工业化的迅速发展、肥料的持续使用和畜牧业的不断发展,导致地球上许多浅水湖泊污染严重,富营养化程度加剧,藻类水华频繁发生,严重影响了湖区生态环境的发展以及人民的生活。浅水湖泊的水华现象也因其频繁性和危害性备受关注。然而,由于众多因素的影响,水华形成的复杂机制至今仍未能得到充分认识。在本论文中,我们希望通过考虑现在关注度较高的一些影响因素,如营养物质、毒素、浮游植物的避难效应、溶解氧等,进一步探索了浅水湖泊水华形成和停止的机制。在此过程中,我们应用到了数学建模以及相应的动力学理论方法的,特别是非线性微分方程的分支理论的应用以及相关计算,其有助于学习湖泊系统的复杂性和解释实际中所观察到的现象。首先,对于两个影响因素,营养物质输入和浮游植物所释放的毒素,早期发现表明:丰富的营养输入会增加藻华形成的几率,而毒素的作用却相反。在第一章,为了研究营养物质输入和毒素对藻华形成的联合影响,我们考虑到毒素对浮游动物捕食能力的影响,从而建立了一个包含营养物质,浮游植物,浮游动物的一般框架模型。通过分析平衡点的存在性,稳定性以及相关的分支现象,我们证明该系统具有复杂的动力学行为,其中包含Hopf分支和余维2 (或更高余维)的Bogdanov-Takens分支。通过选择描述浮游植物营养吸收和浮游动物捕食能力的特殊功能反应函数,我们验证了所获得的理论结果,并进行了相关的数值模拟。我们的结果不仅支持了已有的发现而且还得出系统振荡现象的周期与毒素、营养物质输入水平之间的相互关系:当毒素水平越低或营养素输入量越高时,水华爆发的持续时间将会越长。除了毒素的影响,另外一个引人关注的是浮游植物的避难效应,因为其能够暂时地阻止由于浮游动物捕食所引发的浮游植物灭绝从而使得浮游动植物的生物量处于平稳状态。在第二章里,为了研究毒素和避难效应对浮游动植物之间的动力学行为的影响,我们建立了一个简单的二维模型,其中选择了Holling type Ⅱ的功能反应函数来描述浮游动物对浮游植物的捕食能力。我们给出了相关的理论分析,该系统尽管简单但仍包含了一些复杂的动力学现象,如:Hopf分支和余维2 (或3)的Bogdanov-Takens分支。相关的数值模拟不仅仅支持了我们的理论结果而且在直观上有助于解释所观察到的动力学现象。我们的研究结果表明,浮游植物释放的毒素和能够为浮游植物提供保护的避难所对浅水湖泊藻华的形成和终止具有重要的影响:在某种程度上,它们不仅可以作为水华形成的触发器,而且还可以用于蓝藻水华的生物控制。此外,作为重要的环境影响因素之一的溶解氧不仅对浮游动植物之间的相互作用起到了重要的影响,而且还对藻华的爆发起到了重要的作用。在第三章,考虑到溶解氧对浮游动物捕食能力的影响,我们把浮游植物、浮游动物和溶解氧看作三个单独的变量,建立一个三维的模型来研究溶解氧是如何影响浮游动植物之间的动力学行为,以及如何触发藻华的。相关的理论分析表明该系统能够展示一些复杂的动力学现象,包括:Hopf分支,saddle-node分支和transcritical bifurcation分支。通过选择相关的参数作为分支参数,我们给出了数值模拟,并且对所观察到的动力学现象给出了相应的生物学解释。当其他条件在适当的范围内时,增大溶解氧对浮游动物捕食的最大相对影响率会增加生态系统的不稳定性和水华爆发的风险。而当溶解氧的浓度在适当的范围内时,浮游动物的大量死亡也会增加生态系统的不稳定性和水华爆发的风险。与已有的浮游植物-浮游动物模型相比,除了 hopf分支,当参数选择适当时,我们的系统还会出现双稳态现象,该现象在数学上用来描述在水华过程中藻类生长的特点,同时也说明了该系统中浮游植物和浮游动物共存和最终状态依赖于溶解氧初始浓度。更多还原

【Abstract】 Shallow lake is one of the most important parts of human living environment,which plays a significant role in sustainable development of society and economy. In recent years, as a result of global climate change, rapid industrialization, the growing use of fertilizers and the increasing development of livestock agriculture in the sur-rounding region of lake, many shallow lakes on earth have been seriously polluted,which makes the eutrophication deeper and then the outbreak of algae has become more frequent, which has a crippling effect on the development of the ecological en-vironment and the lives of the people of the lake area. Due to its frequency and harmfulness, algal bloom in shallow lakes has received much more attention recently.However, the complex mechanisms of algal bloom are still far less than understood because of the influences of numerous factors.In this thesis, our aim is to further explore the mechanism of the formation and termination of algal bloom in shallow lakes by considering some factors (mainly in-cluding nutrient, toxin, the refuge for phytoplankton, dissolved oxygen, etc.) which have been paid more attention in recent studies. In this process, mathematical mod-elling and the corresponding theoretical methods of dynamical systems, especially the application and relative calculation of the bifurcation theory of nonlinear differ-ential equations, are very useful to study the complexity of lake ecosystem and give plausible biological explanations for the observed phenomena in reality.Just taking one factor, nutrient input or toxin released by phytoplankton, into consideration, earlier findings show that rich nutrient input can increase the incidence of the bloom-forming but the toxin has the opposite effect. In Chapter 1, we give a general framework for both nutrient uptake by phytoplankton and zooplankton grazing response due to the toxin,using a new nutrient-phytoplankton-zooplankton model to study the combined effect of nutrient input and toxin on algal blooms. We present some of the dynamical analysis of the existence, stability and bifurcations to explore the complex dynamics of the model equations. The dynamics include Hopf bifurcation and Bogdanov-Takens bifurcation of codimension 2 or higher. And then we illustrate our findings by taking specific forms of nutrient uptake and zooplankton grazing with numerical simulations. Except what were discovered in available studies, we also have some more interesting findings on the period of algal blooming and level of toxin, and nutrient input. That is, the duration of the bloom when occurs can be lengthened with the lower toxin level or the higher nutrient input.Besides toxin, another fascinating factor, the refuge provided for phytoplankton,is worthy of attention since it might stabilize the biomass of plankton by prevent-ing extinction of phytoplankton from the predation of zooplankton temporarily. To study the impacts of toxin produced by phytoplankton and refuges provided for phy-toplankton on phytoplankton-zooplankton interactions in lakes,we establish a simple phytoplankton-zooplankton system with Holling type II response function in Chapter 2. The existence and stability of positive equilibria are discussed. Bifurcation analy-ses are given by using normal form theory which reveals reasonably the mechanisms and nonlinear dynamics of the effects of toxin and refuges, including Hopf bifurca-tion, Bogdanov-Takens bifurcation of co-dimension 2 and 3. Numerical simulations are carried out to intuitively support our analytical results and help to explain the ob-served biological behaviors. Our findings finally show that both phytoplankton refuge and toxin have significant impacts on the occurring and terminating of algal blooms in freshwater lakes. To some extent, they may act as not only a biological trigger but also a biological control for algal bloom.Dissolved oxygen, as one of most important environmental factors, plays a sig-nificant role in the interactions between phytoplankton and zooplankton and also in triggering outbreaks of algal blooms. In Chapter 3, considering the effect of dissolved oxygen concentration on the filtering rate of zooplankton, we propose a phytoplankton-zooplankton-dissolved oxygen model to study how dissolved oxygen enriches the dynamical behaviors of plankton ecosystem and triggers algal bloom. We discuss the stability of boundary equilibria, the existence of multiple positive equilib-ria and analyze the local dynamics by studying the bifurcations of a triple equilibrium on a manifold. The system exhibits complex dynamics, including saddle-node bifur-cations, Hopf bifurcation, and transcritical bifurcation. Our theoretical findings are illustrated by performing some numerical simulations with suitable bifurcation pa-rameters, and the corresponding biological interpretations are given for the interesting phenomena from the modeling. With other conditions in suitable range, the maximum relative effect rate of DO on the predation of zooplankton can increase the instabili-ty of ecosystem and the risk of algal bloom. When DO concentration is in a suitable range, the large death of zooplankton can also increase the instability of ecosystem and the risk of algal bloom. Compared with some previous phytoplankton-zooplankton models,besides Hopf bifurcation,the bistable phenomena used in mathematics to de-scribe the dramatic sharp growth and then sudden collapse of phytoplankton during the bloom can appear for suitable parameters. And this bistable phenomena also shows that the coexistence and the final state of phytoplankton and zooplankton depend on initial status of dissolved oxygen concentration.更多还原